Torsional vibration of quartz crystal beams with static axial loads and its application to sensors

This paper describes a vibration analysis of torsional piezoelectric resonators in the presence of axial loads, and the results are applied to sensors, e.g., such as quartz sensors for force, acceleration, and quartz crystal microbalance. A frequency equation for a resonator vibrating in torsion is given as a function of axial load, a thickness-to-width ratio, thickness, and elastic compliance constants. That is, load sensitivity SL (frequency deviation Δf/fo versus axial load F) for this resonator is found to be given as a function of a thickness-to-width ratio and thickness principally, and to be independent of resonator length and the order of vibration. This result is entirely different from that of a flexural mode resonator, and makes it possible to realize miniature piezoelectric devices for sensors